How to use the File Browser's (with ImportHelper) execute function?
Use the ImportHelper - a class that wraps the FileBrowser. 

import os
from bpy_extras.io_utils import ImportHelper

class IdentifierFileSelector(bpy.types.Operator, ImportHelper):
    bl_idname = "something.identifier_file_selector"
    bl_label = "some folder"

    def execute(self, context):
        fdir = self.properties.filepath

        # do something with fdir here  (fullpath)

        return{'FINISHED'}


bl_idname


Should be lowercase. 
For your own sanity it's useful to make it consistent with the class name. If the class name is XxxxxxYyyyZzzzzz then a predictable  bl_idname would be "xxxxx_yyyyy_zzzz"
The prefix (i called it "something" in the code above) is a way for you to create a namespace for your own operators. Each operator is found in bpy.ops.* therefor your operator would be found at bpy.ops.something.identifier_file_selector




In this scenario, the StringProperty isn't something you want to be typing into ever. The update function is a little overkill then. See that property as visual reference to show you which file you picked.

class IgnitProperties(bpy.types.PropertyGroup):
    screen_path = StringProperty(description="Contains path to a texture")


class IdentifierFileSelector(bpy.types.Operator, ImportHelper):
    bl_label = "File Browser"
    bl_idname = "macbook_controller.identifier_file_selector"

    def execute(self, context):
        # Screen Texture
        img = bpy.data.images.load(filepath=self.properties.filepath)

        # update the node
        nt = bpy.data.materials['Screen'].node_tree
        n = nt.nodes["screen_texture"].image = img

        # also update this StringProperty so it reflects what you picked 
        context.scene.ignit_panel.screen_path = self.properties.filepath
        return{'FINISHED'}


your draw

scn = bpy.context.scene
row = layout.row()
row.prop(scn.ignit_panel, "screen_path")  # &lt;-- just for display purposes
row.operator("macbook_controller.identifier_file_selector", text="", icon='FILE')